Mean Field Games theory has been developed since 2006 by J.-M. Lasry and P.-L. Lions to describe Nash equilibria in differential games involving a large population of similar rational agents, where the strategy of each agent depends on the collective behavior. This model is described by parabolic systems of partial differential equations, where a backward Hamilton-Jacobi-Bellman equation is coupled with a forward Kolmogorov-Fokker-Planck equation. This theory has stimulated an increasing interest due to the recent applications in Engineering, Finance and Social Sciences.

In this talk, after a brief introduction to the model, I will present the main peculiarities of these PDE systems and discuss recent advances and open problems about their regularity theory. These led to new questions and answers about qualitative and quantitative properties of solutions of linear and nonlinear partial differential equations.